Dependence modelling and estimation is a key issue in the assessment of portfolio risk. When measuring extreme risk in terms of the Value-at-Risk, the multivariate normal model with linear correlation as its natural dependence measure is by no means an ideal model. We suggest a large class of models and a new dependence function which allows us to capture the complete extreme dependence structure of a portfolio. We also present a simple nonparametric estimation procedure. To show our new method at work we apply it to a financial data set of zero coupon swap rates and estimate the extreme dependence in the data.